July 1981
LABORATORY FOR INFORMATION AND DECISION SYSTEMS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
CAMBRIDGE, MASSACHUSETTS 02139
Status Report on
ESTIMATION AND STATISTICAL
ANALYSIS OF SPATIALLY DISTRIBUTED
RANDOM PROCESSES
NSF Grant ECS-8012668
Covering the Period
From November 1, 1980 to June 30, 1981
Prepared by
Professor Bernard C. Levy
Professor Alan S. Willsky
Submitted to: Dr. A.H. Haddad, Program Director
Systems Theory and Operations Research
Division of Electrical, Computer, and Systems
Engineering
National Science Foundation
Washington, D.C. 20550
LIDS-SR-1114

-2-
SUMMARY
A brief description of the research carried out by faculty, staff, and students of the M.I.T. Department of Electrical Engineering and
Computer Science under NSF Grant ECS-8012668 is presented. The principal investigators for this research are Prof. Bernard C. Levy and Prof. Alan
S. Willsky. The period covered in this status report is November 1, 1980 to June 30, 1981.
The basic scope of this grant is to carry out fundamental research on several interrelated classes of problems involving spatially-distributed random processes. The general philosophy behind our research is to use the structure of the models used to describe spatial processes to obtain insight into the form of solutions and to derive efficient algorithms for their implementation. During the period covered by this status report significant progress has been made in several areas, leading to important results and to promising directions for further research.
The specific topics covered in this report are:
I. The Efficient Sequential Mapping and Analysis of Spatially-
Distributed Data
1.1 Map-Updating and Map-Combining
1.2 Smoothing for Processes Described by Non-Causal Models
1.3 A Scattering Approach to Smoothing and Distributed
Processing
II. Modeling and Extrapolation in Random Fields
III. Detection and Geometric Problems in Random Fields
III.1 Level Crossing Problems for Random Fields
III.2 Detection and Estimation of Objects from Noisy
Projection Measurements
III.3 Efficient Algorithms for Edge Detection
A list of publications supported in whole or in part by this grant is also included.

-3-
I. The Efficient Sequential Mapping and Analysis of Spatially-
Distributed Data
I.1 Map Updating and Map Combining
Significant progress has been made in this area during the period covered by this report. In the original proposal for this research we indicated that problems of map updating and combining had been solved by brute force manipulations for a class of one-dimensional problem formulations suggested by the processing of data obtained from parallel measurement tracks. Much remained to be done both in understanding the results obtained and then using this understanding as a foundation for the investigation of new and more complex problems. In this regard some substantial progress has been achieved.
The results of our work are presented in detail in publications [1] and [3] listed at the end of this report. A brief summary of these results is as follows:
(1) The map updating problem can be viewed naturally as one in which new data is used to estimate errors in a map produced using previously collected data. For the onedimensional problem described above we have succeeded in deriving conditional stochastic realizations; i.e. stochastic dynamic models of map errors. Using these models one can immediately solve the map updating problem, as it reduces to a standard smoothing problem using the map error model. In this way all the results obtained previously in an extremely laborious manner now fall out in a very elegant fashion. In addition, in the process of constructing map error realizations we have developed a unified perspective on smoothing algorithms in general and a simple derivation of map error models.
(2) The extension of our one-dimensional continuous variable results to the discrete parameter case and to the mixed discrete and continuous measurement case has been completed.

-4-
(3) We have made some progress in the use of our results for the processing of two-dimensional maps. In particular, [3] contains a detailed treatment of the application of the map updating and combining algorithms to continuous-parameter random fields with separable covariance functions and track parallel to one of the principal axes. In addition in [3] we also describe an example of the use of our results for a separable discrete-parameter field with non-parallel data tracks. This example represents what we feel is an important breakthrough that indicates that our methodology should be applicable to a wide variety of two-dimensional problems.
(4) We have considered two problems that make use of our map error models in assessing the impact of map error models on the performance of systems. Both of these are related to the level crossing work described in
Section III.1 of this report. In particular in that section we are concerned with the probability that the line of sight between two points on the surface of a randomly rolling terrain is blocked by the terrain.
We have also begun to investigate and have very promising results for the determination of the probability that we will make an error in deciding whether the line of sight is blocked or not, if we base our decision on a map. In addition, we have investigated how map errors can lead to deviations in a desired terrain clearance altitude for a vehicle flying over the terrain. This latter work represents a twist on the problems described in [2].
1.2 Smoothing for Processes Described by Non-Causal Models
Since the physical processes by which random fields are generated are usually noncausal, we have been led to study linear estimation problems for processes described by noncausal models. To do so, we have considered a class of linear noncausal one-dimensional models specified in the form of linear stochastic two-point boundary value systems (TPBVS). This class of models was originally introduced by Krener and it is described either in

differential form by a linear stochastic differential equation with some boundary conditions at both ends of the interval where the process is considered, or in integral form by constructing the Green function associated to the TPBVS. In this case the process is expressed by an integral operator acting on the driving noise process over the whole interval, with an additional term corresponding to the boundary condition contribution. Note that since the models that we consider are noncausal, the processes that we obtain do not have the Markov property in general.
Then, to specify a linear estimation problem, it is assumed that some linear measurements 'of the process are available over the whole interval where the process is defined, as well as some additional measurements at both ends (these are sometimes called post light measurements, since they were originally introduce in flight analysis).
This specifies now a smoothing problem (since the model that we consider is noncausal, the smoothing problem is the only one that can be formulated). The results of our work in this area will be reported in [4].
Our main achievements so far can be described as follows:
(1) The two-filter implementation of the smoother that was originally developed for Markov (i.e. causal) processes can be extended to noncausal processes.
To show this result we have proved that the process generated by the TPBVS can be markovianized, i.e.
it can be embedded in a Markov process of higher dimension. Then the existing smoothing results can be applied to this augmented process. This result is very significant since it shows that even though our original model was noncausal, the smoothing problem can be solved entirely by forwards and backwards causal processing operations. We have also studied the problem of finding the process of minimum dimension required to markovianize our original

-6noncausal process, as well as the associated problem of finding a canonical decomposition of this process into forward and backwards Markovian parts, and into a completely noncausal part.
(2) Another method of solving the noncausal smoothing problem is based on a Hamiltonian formulation of this problem. This requires the construction of a complementary process that contains all the information lost by the original observation process. Then by expanding the original noncausal process with this process, we can express the smoothed estimate as the solution of a two-point boundary value Hamiltonian system. This system, when diagonalized, gives rise to the two-filter smoothing formula mentioned above.
The main advantage of this approach is that it is quite systematic and therefore it could conceivably be applied to more complex (e.g. two-dimensional) noncausal models.
(3) In addition to the previous two solutions, we have also obtained a scattering interpretation of the noncausal smoothing problem of the type that will be discussed in Section 1.3. The main aspects of this interpretation is that the noncausal smoother can be represented by a scattering medium similar to that for the standard (causal) smoother, but with some additional feedback loops between both ends of the scattering medium.
1.3 A Scattering Approach to Smoothing and Distributed Processing
In Section I.1 we have seen that a simple and systematic method of formulating sequential data assimilation problems such as those considered in [1] is to use conditional stochastic realizations. An equally simple and elegant method is to use a scattering theoretic framework. The application of this framework to decentralized estimation problems is discussed in 15]. The main idea is that to each filtering and smoothing problem we can associate an equivalent transmission or scattering problem.

-7-
To solve these problems, instead of using brute-force computational methods, we need only to use pictorial operations which involve: the composition of several scattering layers, the addition of new layers or of new internal sources, the superposition of several layers (provided that the scattering media are the same)...
The media results that we have obtained by using this point of view are as follows:
(1) By associating to some centralized and to some local filtering and smoothing problems some transmission and scattering media, and by using the superposition principle for these media, we have obtained some decentralized filtering and smoothing algorithms.
These algorithms cover both the case when local processors communicate only with a central processor and the case when they communicate between each other.
The map combining and updating problems of Section I.1
can also be analyzed in this framework.
(2) We have also studied the case when the central and local models are different, as well as the case when delays occur in communications between various processors. Finally, we are presently investigating the application of the scattering framework for the solution of estimation problems for weakly coupled or singularly perturbed systems.
II. Modeling and Extrapolation in Random Fields
Our work in this area has had for objective to obtain some efficient extrapolation and interpolation algorithms to estimate a random field given some noisy measurements of this field. Our approach to obtain these algorithms has been to exploit as fully as possible the symmetry properties of measurement geometries as well as the invariance and structural properties of the correlation function describing the random field.
Also, due to the absence of causality in random fields, instead of

attempting to solve causal estimation problems, we have focused our attention on some extrapolation and interpolation problems where no causality assumption is imposed on the process by which measurements are generated. The results of this investigation are described in [6].
The main aspects of this work are as follows:
(1) In the one-dimensional, for a continuous parameter stationary random field, we have considered a symmetric interpolation problem where one wants to estimate the random field at a point given some increasing sets of measurements to the left and right of this point. The solution of this problem depends on the introduction of some even and odd processes obtained by symmetrizing the random field with respect to the point being estimated. These even and odd processes have correlation functions which are the sum of Toeplitz and Hankel kernels, and the solution of our original interpolation problem requires the solution of some even and odd filtering problems associated to the even and odd processes that we have constructed. To solve these filtering problems, we have obtained some vibrating string equations which can also be related to some earlier results of Krein and Dym and McKean on the interpolation and extrapolation of stationary stochastic processes. These string equations provide some recursions for the interpolating filter as the set of available measurement increases. The main feature of these recursions is that they are very efficient, i.e.
the number of operations that they require is quite small by comparison with the number of operations required to recompute entirely the interpolating filter after the measurement set has been expanded.
(2) The results for the symmetric interpolation problem can also be used to solve arbitrary one-dimensional interpolation and extrapolation problems given a finite set of measurements. By doing so we have obtained some new insight on the structure of operators in the algebra generated by Toeplitz and Hankel operators. This insight can be used to obtain some efficient implementations of the solutions of stationary estimation and detection problems. Some similar results also hold for interpolation and extrapolation problems with discrete measurements sets.

-9-
(3) In the two-dimensional case, we have considered the analogous problem of estimating a homogeneous random field given some measurements over a disk (or a doughnut) centered at the point to be estimated.
By expressing the random field in polar coordinates and by expanding it in Fourier series we reduce the interpolation problem to one of estimating the coefficients of the Fourier series expansion. To do so, we use the fact that these coefficients are orthogonal to each other, and that their autocorrelation function can be expressed as the inner product of
Bessel functions. Thus, for each Fourier coefficient process we can associate a filtering problem and by using properties of Bessel functions, we obtain some vibrating membrane equations which can be used to solve our original interpolation problem as well as some more general interpolation and extrapolation problems. These equations are again quite efficient.
In fact for the symmetric interpolation problem the number of operations required is of the same order as for the one-dimensional case.
(4) In the case when the two-dimensional process is only stationary (i.e. its correlation function is invariant under translations but not rotations), we have studied the estimation problem where the measurements are given over a rectangle centered at the point to be estimated.
However, except when the correlation function of the random field is also separable, the algorithms that we have obtained are of higher complexity than in the homogeneous case.
III. Detection and Geometric Problems in Random Fields
III.1 Level Crossing Problems for Random Fields
Significant progress has been made in analyzing one- and two-dimensional level crossing problems described in the original proposal, i.e. problems in which we wish to examine the probability that a line-of-sight exists between two points located at given elevations above a randomly-shaped surface.
These results are described in [3]. Briefly, we have accomplished the following:

-10-
(1) In the one-dimensional case we have derived several different expressions and bounds for the probability of the existence of a line-of-sight and the expected number of crossings of the line-of-sight by the random field. This has been done for two different geometries, corresponding to the original line of sight problem between two given points at fixed elevation, and to the crossing problem associated to a fixed elevation angle line of sight trajectory from a fixed point. These bounds are all expressible in terms of multidimensional integrals of the probability distribution for the random field. A number of these bounds have been calculated numerically for simple
Gaussian process models of the field.
(2) As was mentioned in Section I.1, we have also obtained level crossing bounds for dynamic systems driven by map error fields and bounds on the probability of making an incorrect decision on the existence of a line-ofsight between two points based on the use of a map.
These bounds reduce down to multidimensional integrals which have not yet been examined numerically.
(3) We have also obtained level crossing bounds in two dimensions. These bounds are for the expected number of crossings of a cone by a random field. The cone corresponds to the fixed elevation angle line of sight from a given location rotated through 360°.
These bounds also involve multidimensional integrals for which some numerical results are available.
III.2 Detection and Estimation of Objects from Noisy Projection
Measurements
In this area we have some preliminary results along several directions. These results will be summarized in [7]. Briefly, we have accomplished the following:
(1) We have made some progress in analyzing the solution of an optimal search problem in two-dimensions. In this problem we are looking for the presence of an object in a discrete two-dimensional space. Our possible measurements are sums along rays through the space, where all points except the point where the object is located

contribute independent zero-mean variables to the sum, while the object contributes a known constant.
The optimal measurements strategies for this problem are extremely complex (unlike the one-dimensional case which had been studied earlier by others) and depend on the entire two-dimensional probability distribution for the object location. We have developed some insight into the structure of this solution which we feel we can exploit in deriving optimal and suboptimal measurement strategies.
(2) We have also obtained some promising initial results for the estimation of the location of objects of known shape given a set of line integral measurements.
The form of MAP estimation problems is such that we can exploit the geometry of the problem to obtain useful iterative algorithms. This formulation of the problem is of quite recent vintage (i.e. May 1981) and we feel that it represents an important breakthrough that will allow us to use the geometry of these detection and localization problems to obtain efficient and asymptotically optimal algorithms.
III.3 Efficient Algorithms for Edge Detection
We have very recently initiated a project on the development of extremely efficient algorithms for detecting edges. This is motivated by problems of data compression in which a large number of images are to be trransmitted or stored. While many edge detection techniques have been developed previously, they are in general complex computationally.
This is due to the complexity of the explicit or implicit edge model being used. For problems of fast detection and data compression, simple models must be used to obtain simple algorithms. With this as motivation, we have begun to develop algorithms for detecting straight line edges.
Also, as in some applications one is confronted with a sequence of temporally-related images, we have also begun an investigation of the temporal tracking of straight-line edges.

-12-
PUBLICATIONS
The publications listed below represent papers and reports supported in whole or in part by NSF Grant ECS-8012668.
1. A.S. Willsky, M. Bello, D.A. Castanon, B.C. Levy and G.C. Verghese,
"Combining and Updating of Local Estimates and Regional Maps Along
Sets of One-Dimensional Tracks," revised version in preparation, accepted for publication in the IEEE Trans. on Automatic Control.
2. A.S. Willsky.and N.R. Sandell, "The Stochastic Analysis of Dynamics
Moving Through Random Fields," revised version in preparation; accepted for publication in the IEEE Trans. on Automatic Control.
3. M. Bello, "Centralized and Decentralized Map Updating and Terrain
Masking Analysis," Ph.D. thesis, M.I.T. Department of Electrical
Engineering and Computer Science, to be completed at the end of
July 1981.
4. M. Adams, "Linear Estimation for Noncausal One and Two-Dimensional
Systems," Ph.D. thesis proposal, in preparation.
5. B.C. Levy, D.A. Castanon, G.C. Verghese and A.S. Willsky,
"A Scattering Framework for Decentralized Estimation Problems," to appear in Proc. 20th IEEE Conf. on Decision and Control, Dec. 1981.
6. B.C. Levy and J.N. Tsitsiklis, "Vibrating Strings and Membranes, and the Solution of One and Two-Dimensional Estimation Problems," LIDS
Report, in preparation.
7. D. Rossi, "Detection and Estimation of Objects from Noisy Projection
Measurements," Ph.D. thesis proposal, in preparation.